Equivariant Fredholm modules for the full quantum flag manifold of $SU_q(3)$

We introduce $C^*$-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct $SL_q(3,\mathbb{C})$-equivariant Fredholm modules for the full quantum flag manifold $X_q = SU_q(3)/T$ of $SU_q(3)$, based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold $X_q$ satisfies Poincar\'e duality in equivariant $KK$-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to $SU_q(3)$.